Kal Naismith, who was subject of a bid from Bury earlier this season, scored the winning goal eight minutes from time to move Stanley off the bottom of the League Two table.
Last season they went 12 league games without collecting a win and it had looked like this season could go the same way.
"I am delighted and it is a relief," said Beattie.
"It was a great game to win, as any is, and to get that first win of the season is very pleasing."
Stanley went in front just seven minutes in when James Gray stuck a penalty past debutant goalkeeper Peter Brezovan after Janoi Donacien clipped Naismith in the area.
The visitors fought back, though, and Abdulai Bell-Baggie's brace turned the game on its head. However, with 12 minutes to go, Gray levelled before Naismith popped up for the winner moments later.
Beattie continued: "Tranmere are a good side but our resilience and passion for the badge shone through and that's why we managed to get three points.
"We have started well and scored early but then Tranmere came back. Our heads could have dropped but the two strikers - Marcus Carver and James Gray - have got us through a flat period and helped us win the game.
"Kal had taken a knock just before the winner but I told him to just get in the box and see what he could do and thankfully he has had a nice tap in which I certainly enjoyed.
"It was three points which were richly deserved."
Tranmere boss Rob Edwards was gutted after seeing his team throw away a 2-1 lead in the dying stages.
He said: "We've had so many leads this season and chucked them away; it's driving me mad. I'm as frustrated as anyone.
"We didn't play well in the first half; we didn't do what I wanted the players to do. I didn't want them to just come here, side-foot the ball around and think they were playing well. They thought they were playing well in the first half but we didn't have a shot and Accrington controlled the game.
"We didn't defend our penalty box well enough and have given two goals away. We've got to make better decisions. At times we were anxious and I don't understand why.
"The players have got to hold their nerve and make the right decisions and I thought there were some strange decisions at the back at times. We're all learning the hard way at the minute."
